458 ON METHODS FOR DETECTING SMALL OPTICAL RETAKDATIONS, [415
as the condition that the half maximum brightness of the prism shall coincide with approximate extinction of the remainder of the field of view. If the linear aperture of the prism be b, supposed to be small in comparison with a, the maximum retardation due to it is
O-l)i& = X.6/a;..............................(2)
and we recognize that easy visibility of the prism on the darkened field is consistent with a maximum retardation which is a small fraction of \.
In Cheshire's application of Foucault's method (for I think it should be named after him) the prism had an angle i of 10°, and the aperture a was S cms., although it would appear from the sketch that the whole of it was not used. Thus in (1) \/ia would be about 5 x 10~5; and the accuracy with which /u, was determined (about + '00002) is of the order that might be expected.
It is of interest to trace further and more generally what the wave theory has to tell us, still supposing that the source of light is from an infinitely narrow slit (or, what comes to the same, a slit of finite width at an infinite distance), and that the apertures are rectangular. The problem may then be supposed to be in two dimensions*, although in strictness this requires that the elementary sources distributed uniformly along the length of the slit should be all in one phase. The calculation makes the usual assumption, which cannot be strictly true, that the effect of a screen is merely to stop those parts of the wave which impinge upon it, without influencing the neighbouring parts. In fig. 1, A represents the lens with its rectangular
Fig. i.
aperture, which" brings parallel rays to a focus. In the focal plane B are two adjustable screens with vertical edges, and immediately behind is the eye or objective of a small telescope. The rays from the various points Q of the second- aperture, which unite at a point in the focal plane of the telescope, or of the retina, may be regarded as a parallel pencil inclined to the axis at
* Compare "Wave Theory," Encyc. Brit, 1888 ; Scientific Papers, Vol. in. p. 84.ry diffraction theory the first darkness occurs when the difference of retardations of the various secondary rays issuing from the aperture ranges over one complete wave-length, i.e. when the projection of the aperture on the central secondary ray is equal to X. The half-width (f) of the central band is therefore expressed by f =/X/a.